\chapter*{Abstract}

In metric spaces in general and in $\R^n$ in particular, seguences are often used as a way of characterizing various properties of a topological space. This is founded in the fact that these spaces satisfy the first axiom of countability. In general, however, this is not the case and a more general concept is needed for these characterizations. This is why the theory of net and filters have been developed.

This paper is a thorough examination of the characetization and use of nets and filters. These two concepts are introduced along with various lemmas to aid in the theorems and examples that follow. Although the concepts have been developed seperately, they are equivalent and this will be proved.

Thereafter the generalized theorems from $\R^n$ for closure, continuity, Hausdorff and compactness are proven. These theorems will be presented using both nets and filters and an example of the shifting between use of nets and filters is used as a part of the prove for each theorem. 

Finally several uses of nets and filters are presented. First as a very short proof for Tychonoffs theorem, again using both nets and filters. This is followed with two theorems examplifying that the use of sequences in a topological space satisisfying the first axiom of countability is sufficient. The paper is concluded with two counterexamples of topological spaces in which sequences fail to characterize compactness and the closure of a set.